Advanced Topics - Wellcome Trust Centre for Neuroimaging

Dynamic Causal Modelling
Advanced Topics
SPM Course (fMRI), May 2015
Peter Zeidman
Wellcome Trust Centre for Neuroimaging
University College London
The system of interest
Experimental Stimulus
(Hidden) Neural Activity
Observations (BOLD)
Vector y
on
?
BOLD
Vector u
off
time
time
Stimulus from Buchel and Friston, 1997
Brain by Dierk Schaefer, Flickr, CC 2.0
DCM Framework
Experimental
Stimulus (u)
Neural Model
Observation Model
How brain
activity z
changes over
time
What we would
see in the
scanner, y,
given the
neural model?
.
z = f(z,u,θn)
Observations (y)
y = g(z, θh)
Stimulus from Buchel and Friston, 1997
Figure 3 from Friston et al., Neuroimage, 2003
Brain by Dierk Schaefer, Flickr, CC 2.0
DCM Framework
Experimental
Stimulus (u)
Neural Model
Observation Model
Model Inversion
(Variational EM)
1. Gives parameter estimates:
Given our observations y, and stimuli u, what parameters θ make the
model best fit the data?
2. Gives an approximation to the log model evidence:
log 𝑝(𝑦|𝑚) ≅ Free energy = accuracy - complexity
Observations (y)
Contents
• Creating models
– Preparing data for DCM
• Inference over models
– Fixed Effects
– Random Effects
• Inference over parameters
– Bayesian Model Averaging
• Example
• DCM Extensions
PREPARING DATA
Choosing Regions of Interest
We generally start with SPM results
12
10
8
6
4
2
0
A factorial design translates easily to DCM
Main effect of face: FFA
A (fictitious!) example of a 2x2 design:
Factor 1:
Stimulus (face or inverted face)
Factor 2:
Valence (neutral or angry)
Interaction of
Stimulus x Valence:
Amygdala
Valence
From a factorial design:
Main effects → driving inputs
Interactions → modulatory inputs
Face
FFA
Amy
ROI Options
1. A sphere with given radius
Positioned at the group peak
or
Allowed to vary for each
subject, within a radius of the
group peak
+
2. An anatomical mask
Pre-processing
1. Regress out nuisance effects
(anything not specified in the
‘effects of interest f-contrast’)
2. Remove confounds such as low
frequency drift
3. Summarise the ROI by
performing PCA and retaining the
first component
1st eigenvariate: test
3
New in SPM12: VOI_xx_eigen.nii
(When using the batch only)
2
1
0
-1
-2
-3
-4
200
400 600 800 1000
time \{seconds\}
230 voxels in VOI from mask VOI_test_mask.nii
Variance: 81.66%
Interim Summary: Preparing Data
• DCM helps us to explain the coupling between regions showing an
experimental effect
• We can choose our Regions of Interest (ROIs) from any series of contrasts
in our GLM
• We use Principle Components Analysis (PCA) to summarise the voxels in
each ROI as a single timeseries
Contents
• Creating models
– Preparing data for DCM
• Inference over models
– Fixed Effects
– Random Effects
• Inference over parameters
– Bayesian Model Averaging
• Example
• DCM Extensions
INFERENCE OVER MODELS
Experimental
Stimulus (u)
Neural Model
Observation Model
Observations (y)
Model 1:
Model comparison: Which model best explains my observed data?
Experimental
Stimulus (u)
Model 2:
Neural Model
Observation Model
Observations (y)
Bayes Factor
How good is model 𝑗 versus model 𝑖 in one subject?
𝑝(𝑦|𝑚 = 𝑗)
𝐵𝑗𝑖 =
𝑝(𝑦|𝑚 = 𝑖)
In practice we subtract the log model evidence:
log 𝐵𝑗𝑖 = log 𝑝(𝑦|𝑚 = 𝑗) − log 𝑝(𝑦|𝑚 = 𝑖)
≅ 𝐹𝑗 − 𝐹𝑖
At the group level, over K subjects:
𝑘
𝐵𝑖𝑗
𝐺𝐵𝐹𝑗𝑖 =
𝑘
Bayes Factor - Interpretation
A log Bayes factor of 3 is termed
“strong evidence”.
exp(3) = 20 x stronger evidence for model 𝑗 the than
model 𝑖
Kass and Raftery, JASA, 1995.
We can convert the Bayes Factor to a
posterior probability using a sigmoid
function
BF of 3 = 95% probability
Will Penny
Fixed Effects Bayesian Model Selection (BMS)
Bayesian Model Selection: FFX
35
Log-evidence (relative)
30
25
20
15
10
5
0
1
2
3
4
5
6
7
8
9 10 11 12
Models
Bayesian Model Selection: FFX
Assumption: Subjects’ data arose from
the same underlying model
Model Posterior Probability
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
Models
9 10 11 12
Random Effects Bayesian Model Selection (BMS)
11 out of 12 subjects favour model 2
But… GBF = 15 in favour of model 1
Stephan et al. 2009, NeuroImage
Random Effects Bayesian Model Selection (BMS)
SPM estimates a hierarchical model with variables:
𝑟
𝑚𝑖
𝑦𝑖
The frequency of each model in the group
The model assigned to subject 𝑖
The data for subject 𝑖
Outputs:
𝐸 𝑟2 𝑌 = 0.84
Expected probability of model 2
𝑝 𝑟2 > 𝑟1 𝑌 = 0.99
Exceedance probability of model 2
Stephan et al. 2009, NeuroImage
Random Effects Bayesian Model Selection (BMS)
0.8
Bayesian Model Selection: RFX
Model Expected Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
Models
Assumption: Subjects’ data arose from
different models
Model Exceedance Probability
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Stephan et al. 2009, NeuroImage
Bayesian Model Selection: RFX
1
2
Models
Random Effects Bayesian Model Selection (BMS)
Bayesian Omnibus Risk (BOR)
The probability that all models
have the same frequency in the
group:
BOR = 𝑝0 = 𝑝(𝐻0 |𝑌)
Prob of Equal Model Frequencies (BOR) = 0.45
0.8
Protected Exceedance Probability
Exceedance probability (xp)
assumes each model has a
different frequency in the group. It
is over-confident.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
The xp is adjusted using the BOR
to give the protected exceedance
probability (pxp).
Rigoux et al. 2014, NeuroImage
1
2
Models
Family Inference
Bayesian Model Selection: RFX
Rather than having one hypothesis
per model, we can group models
into families.
E.g. we have 9 models which all
have a bottom-up connection and 5
models which all have a top-down
connection.
Model Expected Probability
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Models
Family Inference
Family Expected Probability
0.8
Bayesian Model Selection: RFX
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Family Exceedance Probability
1
Bottom-up
Families
Top-down
Bayesian Model Selection: RFX
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Bottom-up
Families
Top-down
Interim Summary: Inference over Models
• We embody each of our hypotheses as a model or as a family of models.
• We can compare models or families using a fixed effects analysis, only if
we believe that all subjects have the same underlying model
• Otherwise we use a random effects analysis and report the protected
exceedance probability and the Bayesian Omnibus Risk.
Contents
• Creating models
– Preparing data for DCM
• Inference over models
– Fixed Effects
– Random Effects
• Inference over parameters
– Bayesian Model Averaging
• Example
• DCM Extensions
INFERENCE OVER
PARAMETERS
Parameter Estimates
The estimated parameters for a single model can be found in each
DCM.mat file
Inspect via the GUI
Inspect via Matlab
Estimated parameter means: DCM.Ep
Estimated parameter variance: DCM.Cp
Bayesian Model Averaging
Model Expected Probability
0.35
Bayesian Model Selection: RFX
What are the posterior parameter estimates
for the winning family?
0.3
0.25
0.2
SPM calculates a weighted average of the
parameters over models:
0.15
0.1
𝑝 𝜃𝑦 =
0.05
0
Family Exceedance Probability
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
𝑚
1 2 3 4 5 6 7 8 9 1011 1213 14
Models
𝑝(𝜃, 𝑚|𝑦)
=
𝑝 𝜃 𝑚, 𝑦 𝑝(𝑚|𝑦)
𝑚
Region 1 to 2
To give a posterior
distribution for each
connection:
Bottom-up
Top-down
Families
Contents
• Creating models
– Preparing data for DCM
• Inference over models
– Fixed Effects
– Random Effects
• Inference over parameters
– Bayesian Model Averaging
• Example
• DCM Extensions
EXAMPLE
Reading > fixation (29 controls)
Lesion (Patient AH)
1. Extracted regions of interest. Spheres placed at
the peak SPM coordinates from two contrasts:
A. Reading in patient > controls
B. Reading in controls
2. Asked which region should receive the driving
input
3. Performed fixed effects BMS in the patient and
random effects BMS in the controls, and applied
Bayesian Model Averaging.
Bayesian Model Averaging
Key:
Controls
Patient
Seghier et al., Neuropsychologia, 2012
Seghier et al., Neuropsychologia, 2012
Interim Summary: Example
• When we don’t know how to model something, we can ‘ask the data’ using
a model comparison
• Bayesian Model Averaging (BMA) lets us compare connections across
patients and controls
• Posterior parameter estimates can be used as summary statistics for
further analyses
DCM EXTENSIONS
DCM Framework
Experimental
Stimulus (u)
Neural Model
Observation Model
How brain
activity z
changes over
time
What we would
see in the
scanner, y,
given the
neural model?
.
z = f(z,u,θn)
Observations (y)
y = g(z, θh)
Stimulus from Buchel and Friston, 1997
Figure 3 from Friston et al., Neuroimage, 2003
Brain by Dierk Schaefer, Flickr, CC 2.0
DCM Extensions
Non-Linear DCM
Two-State DCM
modulation
driving
input
Stephan et al. 2008,
NeuroImage
Marreiros et al. 2008,
NeuroImage
DCM Extensions
Stochastic DCM
Li et al. 2011,
NeuroImage
DCM for CSD
Friston et al. 2014,
NeuroImage
DCM Extensions
Post-hoc DCM
Friston and Penny 2011,
NeuroImage
Further Reading
The original DCM paper
Friston et al. 2003, NeuroImage
Descriptive / tutorial papers
Role of General Systems Theory
Stephan 2004, J Anatomy
DCM: Ten simple rules for the clinician
Kahan et al. 2013, NeuroImage
Ten Simple Rules for DCM
Stephan et al. 2010, NeuroImage
DCM Extensions
Two-state DCM
Marreiros et al. 2008, NeuroImage
Non-linear DCM
Stephan et al. 2008, NeuroImage
Stochastic DCM
Li et al. 2011, NeuroImage
Friston et al. 2011, NeuroImage
Daunizeau et al. 2012, Front Comput
Neurosci
Post-hoc DCM
Friston and Penny, 2011, NeuroImage
Rosa and Friston, 2012, J Neuro Methods
A DCM for Resting State fMRI
Friston et al., 2014, NeuroImage
EXTRAS
Variational Bayes
Approximates:
The log model evidence:
Posterior over parameters:
The log model evidence is decomposed:
The difference between the true and
approximate posterior
Free energy (Laplace approximation)
Accuracy
-
Complexity
The Free Energy
Accuracy
-
Complexity
Complexity
Distance between
prior and posterior
means
Occam’s factor
Volume of
prior parameters
posterior-prior
parameter means
Prior precisions
(Terms for hyperparameters not shown)
Volume of
posterior parameters
DCM parameters = rate constants
Integration of a first-order linear differential equation gives an
exponential function:
dx
 ax
dt
x ( t )  x 0 exp( at )
Coupling parameter a is inversely
proportional to the half life  of z(t):
x ( )  0.5 x 0
The coupling parameter a
thus describes the speed of
the exponential change in x(t)
0.5 x 0
 x 0 exp( a )
a  ln 2 / 
  ln 2 / a
Practical Workshop
Attention to Motion in the visual system
DCM – AttentionStimuli
to Motion
250 radially moving dots at 4.7 degrees/s
Paradigm
Pre-Scanning
5 x 30s trials with 5 speed changes (reducing to 1%)
Task - detect change in radial velocity
Scanning (no speed changes)
F A F N F A F N S ….
F - fixation
S - observe static dots
N - observe moving dots
A - attend moving dots
Parameters
- blocks of 10 scans
- 360 scans total
- TR = 3.22 seconds
+ photic
+ motion
+ attention
Slide by Hanneke
den Ouden
Attention to Motion in the visual system
Paradigm
Results
SPC
V3A
V5+
Attention – No attention
Büchel & Friston 1997, Cereb. Cortex
Büchel et al. 1998, Brain
- fixation only
- observe static dots
- observe moving dots
- task on moving dots
+ photic
+ motion
+ attention
 V1
 V5
 V5 + parietal cortex
Slide by Hanneke
den Ouden
DCM: comparison of 2 models
Model 1
attentional modulation
of V1→V5: forward
Photic
SPC
V1
Model 2
attentional modulation
of SPC→V5: backward
Photic
Attention
V1
V5
Motion
Attention
SPC
Motion
V5
Bayesian model selection: Which model is optimal?
Slide by Hanneke
den Ouden
Attention to Motion in the visual system
DCM – GUI basic steps
1 – Extract the time series (from all regions of interest)
2 – Specify the model
3 – Estimate the model
4 – Review the estimated model
5 – Repeat steps 2 and 3 for all models in model space
6 – Compare models
Slide by Hanneke
den Ouden

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